Galileo's example of the set of all squares of positive integers
and the set of all positive integers:
1*, 32, 32, 42, ... ,r*2, ...
1 ,2,3,4, ... ,7i,...
The 'paradoxical' distinction between this and the preceding
examples is apparent. If all the wives retire to the drawing
room, leaving their spouses to sip port and tell stories, there will
be precisely twenty human beings sitting at the table, just half
as many as there were before. But if all the squares desert the
natural numbers, there are just as many left as there were
before. Dislike it or not as we may (we should not, if we are
rational animals), the crude miracle stares us in the face that a
part of a set may have the same cardinal number as the entire set.
If anyone dislikes the 'pairing' definition of 'same cardinal
number', he may be challenged to produce a comelier. Intuition
(male, female, or mathematical) has been greatly overrated.
Intuition is the root of all superstition.
Notice at this stage that a difficulty of the first magnitude
has been glossed. What is a set, or a class? 'That', in the words of
Hamlet, 'is the question*. We shall return to it, but we shall not
answer it. "Whoever succeeds in answering that innocent ques-
tion to the entire satisfaction of Cantor's critics will quite likely
dispose of the more serious objections against his ingenious
theory of the infinite and at the same time establish mathe-
matical analysis on a non-emotional basis. To see that the
difficulty is not trivial, try to imagine the set of all positive
rational integers, I, 2, 3, *.. , and ask yourself whether, with
Cantor, you can hold this totality - which is a 'class' - in your
mind as a definite object of thought, as easily apprehended as
the class x, y, z of three letters. Cantor requires us to do just
this thing in order to reach the transflnite numbers which he
Proceeding now to the definition of 'cardinal number"1, we
introduce a convenient technical term: two sets or classes
whose members can be paired off one-to-one (as in the examples
given previously) are said to be similar. Haas many things are
there in the set (or class) x, y, s? Obviously three. But what is